Xiao; General age- and time-dependent growth models for animals 



699 



to be calculated), and a reference time / ' (known). 



r 



Both reference times must be chosen properly, such 

 that t^=t . on ^^.'s scale and t ^_'=t, on /^,"s scale, where t . 

 is an arbitrarily chosen time. For example, /,=0 on 

 t 's scale; ^'=0 on / "s scale. Projection of both t =t , 

 on t 's scale and t/= t . on ^^."s scale onto a third time 

 scale (projection time scale. Fig. 1) to find their time 

 difference on the third time scale z'-r... It is this time 



difference that is to be used to calculate t^^'. To do so. 



fo=f'- 



let t-t^=t '-t ', or t'=tM '-t =t.+ T'-T J 



r (p r ' r r r /' 



tirt^'+t=t'-t^-T^'+T^. Therefore, in Equations 6.2, 10.2, 

 and 14.2, replacement of f-t^ with t'-t^-T/+z^., of a- 

 Oq with a'-flg', and of f-t^^ with ^'-^q' will give 

 the correct growth models for the future fish 

 stock assessment on the required time scale 

 (application scale). For the barramundi 

 growth described by equation 6.2, t^ = 

 62.7197 d, /,,=0 corresponds to r,.= l Janu- 

 ary 1960 (row 2, Table 3), /,,'=0 corresponds 

 to r/=l January 1999, then t^'-t +t/- 

 t^=62.7197+(l January 1999 )-( January 1, 

 1960)^62. 7 197 + 14245^:14307. 7 197 d. 

 Therefore, replacement of t—t with t'- 



Application time scale 

 Projection time scale 

 Regression time scale 



/ =/. 



t=l. 



Figure 1 



Relation among regression, application, and projection time 

 scales for adjusting estimates of parameters in Equations 

 6.2. 10.2, and 14.2 for subsequent applications. 



and of t-t, 







14307.7197, of a-a„ with a'-a^' 

 with t'-tQ in the model concerned will give 

 the correct growth models for the future 

 (when time starts on 1 January 1999) bar- 

 ramundi stock assessment on the required 

 time scale (application scale). In this ex- 

 ample, the third time scale (projection time 

 scale) is, of course, calendar time. 



Discussion 



This work presents general age- and time- 

 dependent models for the growth of animals 

 and a comprehensive list of their useful special cases, 

 forming a basis for obtaining quantitative informa- 

 tion on the growth of animals experiencing changes 

 in age, time, and age- and time-varying factors. These 

 models have many applications. An obvious one 

 would be to examine both the short- and long-term 

 effects of tagging on the growth of animals by use of 

 Equations 6.3, 10.3, and 14.3; iiC <K ,K =K , 



^ ' ' ' max nun' max mm' 



andK^^^^^_>K^^^^^^ indicate, respectively, positive, no, and 

 negative effects of tagging on the growth of animals. 

 Similarly, i^,„„ <0, /f,„„,=0, and if„„„>0 suggest, re- 

 spectively, a shrinkage, cessation of growth, and a 

 slower growth of tagged animals immediately after 

 tagging. In the case of the L. calcarifer (Fig. 2), tag- 

 ging seems to have been antagonistic to its growth 

 (A'„„ ,>X, ,, ) and led to a shrinkage of its size iK <0). 



n\ax nun ^ nun 



This conclusion is tentative, however, because of the 

 large standard error of d . 



Another application would be to study how age- 

 and time-dependent factors other than age and time 

 affect the growth of animals. For example, one can 

 hypothesize about the functional forms of KUi.t), such 

 as K(a,t)=a.T(t-t^J^, where T(t) is ambient tempera- 

 ture, availability of food, or pH value; t^. (a time lag 

 or lead), and a and [3 are all parameters to be esti- 

 mated or specified. Such a model is ideal for analyz- 

 ing data on the length or weight of an individual 

 animal at age and time, which may be available, say, 

 from aquaculture operations. It might also be useful 

 for analyzing data from mark-recapture experiments, 

 where ambient temperature or food availability of a 

 tagged animal is measured continuously from the 

 time of its tagging to the time of its recapture. In- 

 deed, if L. calcarifer had been tagged with a "smart" 

 tag that could record ambient temperature or food 

 availability, analysis would have been made of their 



